John Todd and the Development of modern Numerical Analysis

Irish Math. Soc. Bulletin
Number 69, Summer 2012, 11–23
ISSN 0791-5578
JOHN TODD AND THE DEVELOPMENT OF
MODERN NUMERICAL ANALYSIS
NIALL MADDEN
Abstract. The purpose of this article is to mark the centenary of
the birth of John Todd, a pioneer in the fields of numerical analysis
and computational science. A brief account is given of his early life
and career, and that of his wife, Olga Taussky, including experiences during World War II that led to him engaging with the then
developing field of numerical analysis. Some of his contributions to
the field, and the contexts in which they arose, are described.
1. Before the war
John (Jack) Todd’s long and eventful life began on May 17th, 1911
in Carnacally, County Down. I give only an outline of these events
here, and refer the interested reader to [1, 2, 5] for further details.
Having attended Methodist College in Belfast, Todd studied at
Queen’s University Belfast from 1928 to 1931, where A.C. Dixon was
professor of Mathematics. He then went to Cambridge, but could
not enrol for a bachelor’s degree since he had not studied Latin, and
so became a research student instead. He was supervised by J.E.
Littlewood, who disapproved of the notion of doctoral degrees, so
Todd never completed one. He worked under Littlewood’s guidance
on transfinite superpositions of absolutely continuous functions [25,
26].
He returned to lecture in Queen’s University Belfast in 1933, working with J.G. Semple who had recently been appointed as professor.
When Semple moved to King’s College London in 1937, he invited
2010 Mathematics Subject Classification. 01-08, 01A60, 10A70, 65-XX.
Key words and phrases. John Todd, Olga Taussky, Biography, Numerical Analysis, Computational Mathematics.
Received on 14-5-2012; revised 29-5-2012.
The author is grateful to Prof. Tom Laffey for discussions about Olga Taussky
and John Todd, to Frank Uhlig helpful comments and for the image in Figure 2,
and to the Archives of the Mathematisches Forschungsinstitut Oberwolfach for
permission to reproduce images in Figure 1.
c
2012
Irish Mathematical Society
11
12
N. MADDEN
Todd to join him. Initially Todd taught aspects of real analysis
there, particularly measure theory, but when another professor became ill while teaching a course in group theory, he was asked to
take over.
He developed an interest in the area — enough to attend seminars
on the topic at other London Colleges, and tackle a challenging
research problem. That in turn led him to contact Olga Taussky
who was at Westfield College, leading to a personal and professional
partnership that was to last for nearly 60 years.
Figure 1. John Todd, 1977 and Olga Taussky, (circa
1932).
(Source: Archives of the Mathematisches
Forschungsinstitut Oberwolfach)
2. Taussky and Todd
Olga Taussky was a highly prolific and influential mathematician:
she authored roughly 200 research articles and supervised the research of 14 graduate students; she was a founding editor of the
journals Linear Algebra and its Applications and Linear and Multilinear Algebra; she received many awards and distinctions, including
election as vice-president of the American Mathematical Society in
1985.
Taussky was born in 1906 in Olmütz, in the Austro-Hungarian
empire. She studied in Vienna, initially majoring in both mathematics and chemistry, the latter being related to the family business.
JOHN TODD
13
However, she soon focused her attention on mathematics, eventually
completing a doctorate in Vienna, under the supervision of Philip
Furtwängler, a noted number theorist who contributed significantly
to the development of class field theory.
In 1931 she moved to Göttingen, primarily to work with Richard
Courant on editing Hilbert’s papers on number theory, while also assisting Courant and Emmy Noether with their courses. However, the
rise of antisemitism resulted in many academics in German universities, including Courant, Noether and Taussky, being forced to leave
their positions. After a short time in Cambridge, Courant became a
professor at New York University in 1936, while Noether moved to
Bryn Mawr College in Pennsylvania. Taussky was awarded a three
year research fellowship from Girton College in 1935, and decided
to spend the first year of that in Bryn Mawr with Noether.
After applying for numerous positions, Olga Taussky was eventually appointed to a teaching post at Westfield College, a constituent
of the University of London. In 1937 she met Todd and within a
year they married, somewhat inauspiciously, on the day the Munich Agreement was signed. Their first joint articles, which show
Taussky’s emerging interest in the developing field of matrix theory
(e.g., [21, 22]) were written in a bomb shelter in London. Their
final joint paper [23], a historical note on links between the celebrated method of Cholesky and work of Otto Toeplitz, was completed shortly before Taussky’s death in 1995.1
For more details of Taussky’s life and career, see [17] and [47],
and the autobiographical articles [24] and [20]: the former is primarily concerned with her life and experiences, the latter with her
contributions to matrix theory. Her contributions to other areas of
algebra are discussed in [16].
3. During the war
With the outbreak of World War II, and their colleges evacuated
from London, Taussky and Todd moved to Belfast where they both
1
The manuscript was originally submitted in 1995. Following the death of
Taussky later that year, the manuscript was “lost” for several years. Some years
after, there was renewed interest in the origins of Cholesky’s method, including
the discovery by Claude Brezinski of an original, unpublished, hand-written note
by Cholesky describing it. So in 2005, Todd resubmitted the paper for publication
in Numerical Algorithms.
14
N. MADDEN
taught for a year. Eventually they were to return to London and to
work in scientific war jobs.
Taussky worked on aerodynamics at the National Physics Laboratory with the Ministry of Aircraft Production. Here she learned
a great deal about differential equations, which had not interested
her much previously, and matrix theory. She worked on problems in
flutter [44], expressed as hyperbolic differential equations, and developed a technique that greatly reduced the amount of computational
effort required to estimate the eigenvalues of certain matrices. Her
idea was to use the simple, but very powerful, idea introduced by
Gers̆gorin [12] which shows that the eigenvalues of an n × n matrix
A with complex entries are contained in theP
union of n disks, where
th
the i disk is centred on aii and has radius j6=i |aij |. Taussky then
used carefully chosen similarity transforms that reduce the radius of
the disks, thus improving the accuracy of the estimate.
Although Gers̆gorin’s work had received some attention, Taussky
is often credited with popularising it, for example in [19]. Many
generalisations and extensions were to follow, by Taussky and by
others; an accessible account of these is given by Varga in [48]. The
study of Gers̆gorin’s disks were also a topic of research in the Ph.D.
studies of Taussky’s Irish student, Fergus Gaines [11].
While Taussky was working on aeronautics, Todd worked with the
Admiralty in Portsmouth, initially on ways of counteracting acoustic
mines. During that period he was struck by the amount of time that
physicists spent doing routine calculations, while mathematicians
were attempting to engage with engineering problems:
“This was rather frustrating: physicists were doing
elementary computing badly and mathematicians like
me were trying to do physics. I thought that I could
see a way to improve this mismatching” [1].
Todd persuaded his superiors to reassign him to London to establish
what became the Admiralty Computing Service, centralising much
of numerical computations for the naval service. He remained there
until 1946.
In 1942 John von Neumann visited the Admiralty to inspect some
of their ballistic facilities in connection with his work on developing the atomic bomb at Los Alamos. Todd was given the task of
accompanying him, and introduced him to their computing facility.
JOHN TODD
15
This led to the rather remarkable claim, made by von Neumann,
that Todd was responsible for getting him interested in computing!
Todd’s work with the Admiralty also led to what he considered
his greatest contribution to mathematics. In 1945 he was part of a
small group that visited Germany to investigate mathematics that
might be of interest to the Navy, such as the work of Konrad Zuse
on programmable computers. The group also visited Oberwolfach,
which they had heard was being used as a mathematics research
centre. They arrived just in time to prevent it from being looted by
Moroccan soldiers. Because it was in the French zone of occupation,
Todd subsequently travelled to Paris to persuade the French government to maintain it. It later developed into a world renowned centre
for mathematical research. For a lively recounting of the adventures
of this time, see [41].
4. Conversion to Numerical Mathematics
Even before setting up the Admiralty Computing Service, Todd
had developed an interest in the topic of computing, initially prompted by Alan Turing’s work on computable numbers [45], and through
contacts with the British Association for the Advancement of Science (BAAS), which was mainly concerned with making tables—
often regarded as the primary goal of early scientific computing.
When he returned to Kings College in 1946, he taught the first
course there on numerical mathematics. There were no text books
for this developing area, so Todd developed his own notes. This
included a section on the solution of systems of linear equations,
featuring the method of Cholesky, which at the time was not well
known in the mathematics literature.2 It was through this course
that Leslie Fox and colleagues at the Mathematics Division of the
National Physics Laboratory became aware of this now standard
method.
2André-Louis
Cholesky was a French military officer. He developed his eponymous method for solving linear systems involving Hermitian, positive-definite matrices when he was engaged in computing solutions to certain least-squares problems that arise in geodesy. Compared to other approaches at the time, it was
remarkably efficient—Cholesky reported that he could solve a system of 10 equations, to 5 decimal digits of accuracy, in under five hours! He explained his method
to other topographers, but never published it. It was eventually published in a
journal on geodesy by a colleague several years after Cholesky’s untimely death
towards the end of World War I [6]. See also [3].
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N. MADDEN
Around this time, Artur Erdélyi and Todd wrote an article in
Nature based on their observations of industrial mathematics research [9]. They argued for the foundation of an Institute for Practical Mathematics in the U.K. to provide instruction in the “mathematical technology” needed for developments in engineering, mathematical biology and economics. The call was not immediately
heeded, but in 1947, Todd and Taussky moved to the United States,
at the invitation of John Curtiss, to help establish the new Institute
for Numerical Analysis at the National Bureau of Standards. Following an “inspirational” three months visiting von Neumann at
Princeton [2], they began working at the INA, located at first at
UCLA. They moved to Washington a year later where they stayed
for 10 years.
In 1957 Todd and Taussky were offered positions at Caltech: John
as Professor of Mathematics, and Olga as a research associate “with
the permission but not the obligation to teach” [17].3 Todd’s appointment was made so as to develop courses in numerical analysis and computation within the mathematics department. They
remained at Caltech for the rest of their lives. Taussky died on
October 7th , 1995. Todd died on May 16th , 2007.
5. Todd and Numerical Analysis
Prior to the 1940s, a “computer” was usually understood to be a
person who carried out calculations, and the designers of numerical
algorithms had in mind the development of methods that were to
be implemented by hand. Computing machinery was mainly limited to hand-operated mechanical calculators, such as the 10-digit
Marchant Model 10 ACT—the first calculator used in Todd’s course
on numerical analysis at King’s College in 1946. The same year,
ENIAC (Electronic Numerical Integrator And Computer ), the first
general-purpose, programmable electronic computer, was launched.
From the 1940s, the rapid development of computer hardware was
mirrored by rapid developments in the field of numerical analysis:
it could be argued that the field emerged as a discipline in its own
right between 1940 and 1960. (The term itself is usually credited to
3At
the time, Caltech regulations prevented Todd and Taussky from holding
professorships concurrently; this was only modified years later as a result of the
recognition that Taussky was receiving as one of the country’s leading mathematicians. Taussky was granted tenure in 1963 and a full professorship in 1971.
JOHN TODD
17
John Curtiss, and its first use was in the name of the Institute for
Numerical Analysis that he founded in 1947). Landmark advances
included the development of the Crank-Nicolson method for timedependent partial differential equations in 1947, and the discovery
of the more computationally efficient alternating direction method
(ADI) by Peaceman and Rachford in 1953. The basis for Finite
Element methods, now the most popular approach for numerical solution of partial differential equations in engineering applications,
was provided by Courant in 1943, but their full potential was not
realised until the 1960s. In 1947, von Neumann and Goldstine produced the first mathematical study of direct numerical solution of
linear systems, with particular regard to the effects of round-off error.4 New algorithms (and their analyses) for the iterative solution
of linear systems included successive over-relaxation (SOR) in 1950,
and the Conjugate Gradients method in 1952 (though the latter
did not achieve significant popularity until much later). The Fast
Fourier Transform of Cooley and Tukey was developed in 1965.
The American Mathematical Society’s journal Mathematical Tables and Other Aids to Computation was launched in 1943, and
renamed Mathematics of Computation in 1960. In 1959, the first
journal for numerical analysis, Springer-Verlag’s Numerische Mathematik was founded, with Robert Sauer, Alwin Walther, Eduard
Stiefel, Alston Householder, and John Todd as editors. Todd served
on the editorial board for 49 years. The SIAM Journal on Numerical
Analysis was founded later, in 1964.
Readers interested in the history of the development of the field
of numerical analysis in the 20th century should consult the recent
article by Grcar [14] which pays special attention of the importance
of von Neumann’s work, particularly [49], and those who developed
its ideas further, including John Todd.
5.1. Articles. Todd’s first papers in numerical analysis were related
to computational linear algebra, and the problems of ill-conditioning
of matrices. Suppose A is a nonsingular matrix and we wish to solve
the system Ax = b by computational means. Simply representing b
with finitely many digits introduces numerical error. In many cases
4Their
paper [49] is often cited is the first in modern numerical analysis; others
would give that honour to Turing for his 1936 paper on computable numbers [45],
while in [38] Todd points to Comrie’s 1946 article on the use of general business
machines in solving computational problems [7].
18
N. MADDEN
of interest, the implementation of standard direct solution algorithm
(all variants on Gaussian elimination) on a computer with finite
precision greatly magnifies these errors. In [49], von Neumann and
Goldstine introduced a quantity they called the “figure of merit”
which gives an upper bound on the magnification of the errors. This
is usually denoted κ(A) and is now known as the condition number of
the matrix, a term coined by Alan Turing in a related work written
around the same time [46]. If κ(A) is large, then the matrix is said
to be ill-conditioned. It is usually defined as, for example,
κ(A) = kAkkA−1 k,
where k·k is one’s favourite matrix norm, or as the ratio of the largest
to smallest eigenvalue of A. This latter case can be useful in practice,
since it does not require direct knowledge of A−1 . Furthermore,
in [49] the “figure of merit” is given as kAk2 kA−1 k2 = σn /σ1 where
0 < σ1 ≤ · · · ≤ σn are the singular values of the invertible matrix A.
But since in most of the cases considered, A is a symmetric positive
definite matrix, this is the same as λn /λ1 where 0 < λ1 < · · · < λn
are the eigenvalues of A. Turing’s proposed measures included using
a scaled Frobenius norm. Todd [27, 29] studied this issue for a matrix
arising in the numerical solution of a second-order elliptic problem in
two variables by the standard finite difference method, and showed
the relationship between several measures of conditioning proposed
by Turing, von Neumann and others. His work was instrumental in
Goldstine and von Neumann’s quantity becoming accepted as the
condition number. He went on to study fourth and higher-order
problems in [32]. See [14] for a further discussion of this.
Other articles, including [28], are concerned with the stability of
finite difference schemes, and propose that such analysis be done
based on the matrix analysis of resulting linear systems. Similar ideas, but for several explicit and implicit schemes for timedependent problems, are found in [34]. A mathematical analysis
(explaining experimental results obtained by other authors) for an
ADI method is given in [10]. The computation of special functions
features in [15], for example, and the efficiency of methods for solving integral equations is considered in [35].
Although numerical analysis is often (and certainly, originally) understood as the mathematical study of computer algorithms for solving mathematical problems, in [33] Todd coined the term “ultramodern numerical analysis” (or “adventures with high speed automatic
JOHN TODD
19
digital computing machines”) which, unlike other areas of mathematics of the time, features aspects of experimentation, particularly
where rigorous error analysis was not possible for sufficiently complicated problems. A systematic study of such experimentation, using
matrix inversion as the main illustration, is given in [18]. An experimental study of a linear solver is given in [30], and of the application
of a Monte Carlo method for solving a partial differential equation
in [31] (as originally proposed by Courant, Friedrichs and Lewy in
their celebrated 1928 paper that includes their famous condition for
the stability of explicit schemes for time dependent problems [8]).
Given the need to avoid over-extrapolation based on numerical experiments for a limited number of examples, Todd [33] cautioned
that “separation of theoretical and applied numerical analysis is undesirable”.
Although many of Todd’s later papers were on the history of computational mathematics, he continued publishing original research
into his seventies [4, 42] and eighties [43].
5.2. Books. While at the Bureau of Standards, Todd developed a
programme to help train mathematicians in the new techniques of
computational mathematics. He arranged for experts in the field to
give courses in particular topics. At the suggestion of Taussky, the
notes from these courses developed into the highly influential Survey
of Numerical Analysis [36]. The first chapter, a reworking of [33]
mentioned above, is titled Motivation for Working in Numerical
Analysis, and notes that
“the profession of numerical analysis is not yet so desirable that it is taken up by choice; indeed, although
it is one of the oldest professions, it is only now becoming respectable”.
He distinguished between what he termed classical, modern and ultramodern numerical analysis. Classical numerical analysis is concerned with solution, by hand, of problems in interpolation, integration, and the approximation of solutions to initial value problems5.
Modern numerical analysis, on the other hand, is required for the
exploitation of automatic digital computation. Finally, ultramodern
numerical analysis relies on a combination of rigorous mathematics,
5See
the fascinating monograph [13] for details on the fundamental developments of classical numerical analysis.
20
N. MADDEN
where problems permit it, and careful study of experimental results
for problems whose complexity is beyond existing mathematical theory (for example, one might devise a numerical method for solving
a nonlinear differential equation, give a mathematical analysis for
a linearised variant and the results of supporting numerical experiments for the full nonlinear problem).
The survey contained contributions from, among others, Morris
Newman, Harvey Cohen, Olga Taussky, Philip J. Davis, Werner
Rheinbolt and Marshall Hall, Jr.. Topics covered include approximation of functions, the principles of programming, Turing Machines
and undecidability, numerical linear algebra, differential equations,
integral equations, functional analysis, block designs, number theory, and computational statistics.
Todd did not completely abandon his earliest research interests.
His 1963 monograph, Introduction to the Constructive Theory of
Functions [37] drew from the tradition of classical analysis to present
sometimes neglected ideas on Chebyshev theory and orthogonal
polynomials, while still providing “some mild propaganda for numerical analysis”.
The courses in numerical analysis that Todd developed at Caltech became the basis for the two volume Basic Numerical Mathematics. As educational aspects of the subject developed, most
presentations where aimed either at students at graduate or upper
undergraduate level, or incorporated computational aspects into introductory courses in linear algebra and calculus. As he explains in
Volume 1 [40], Todd aimed to introduce aspects of numerical computation after only the basics of calculus and algebra had been studied.
He combined both “controlled numerical experiments”, to reinforce
ideas such as convergence and continuity, with “bad examples”, to
temper the tendency to rely on numerical experience rather then
develop sound mathematical analyses:
“The activities of the numerical analyst are similar to
the highway patrol. The numerical analyst tries to
prevent computational catastrophes”.
Often, the existence of such “bad examples” is due to the subtle difference between real numbers and those that might be represented
JOHN TODD
21
by computer. As an example (see [40, Chap. 3]) consider the divergent harmonic series
1 1 1
+ + + ··· .
2 3 4
If this sum is constructed on a computer it will appear to be convergent since, for large enough n, the computer will not be able to
distinguish 1/n from zero.
Volume 1 [40], subtitled “Numerical Analysis”, covers topics in
interpolation, quadrature and difference equations, and are complemented by (relatively) elementary programming exercises. Since
most programs required are for scalar problems, students were expected to develop a complete implementation of the algorithms
themselves.
Volume 2 [39], “Numerical Algebra” deals with direct and iterative
methods for solving linear systems of equations, and for the estimation of eigenvalues, with applications to curve fitting and solution of
boundary value problems. Because the algorithms require the representation and storage of vectors and matrices, unlike the earlier
volume, students were encouraged to use libraries of subroutines to
complete programming assignments.
To summarise the importance, not only of these books, but of
Todd’s contribution in general, we give a quotation from A. S.
Householder’s review of [40]:
“Probably no one has a practical and theoretical background surpassing that of the author, and this book is
altogether unique”.
1+
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38(1):2–23, 2007.
[2] F. L. Bauer. In memoriam John Todd (1911 – 2007). Numer. Math., 108(1):1–
6, 2007.
[3] C. Brezinski. The life and work of André Cholesky. Numerical Algorithms,
43(3):279–288, 2006.
[4] B. Carlson and J. Todd. The degenerating behavior of elliptic functions.
SIAM J. Numer. Anal., 20:1120–1129, 1983.
[5] S. K. Cohen. Interview with John Todd. California Institute of Technology
Archives Oral History Project, 1997.
[6] Commandant Benoit. Note sur une méthode de réolution des équations normales provenant de l’application de la méthode des moindres carrés á un
22
N. MADDEN
Figure 2. John Todd and Olga Taussky-Todd, March
1973, Los Angeles. (Source: Frank Uhlig)
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Accounting Machine. J. Roy. Statist. Soc., 3(2):87–114, 1936.
R. Courant, K. Friedrichs, and H. Lewy. Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann., 100:32–74, 1928.
A. Erdélyi and J. Todd. Advanced Instruction in Practical Mathematics.
Nature, 158:690–692, 1946.
D. Gaier and J. Todd. On the rate of convergence of optimal ADI processes.
Numer. Math., 9:452–459, 1966/1967.
F. Gaines. Some Generalizations of Commutativity for Linear Transformations on a Finite Dimensional Vector Space. PhD thesis, California Institute
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JOHN TODD
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[16] T. J. Laffey. Some aspects of Olga Taussky’s work in algebra. Linear Algebra
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[18] M. Newman and J. Todd. The evaluation of matrix inversion programs. J.
Soc. Ind. Appl. Math., 6:466–476, 1958.
[19] O. Taussky. A recurring theorem on determinants. Amer. Math. Monthly,
56:672–676, 1949.
[20] O. Taussky. How I became a torchbearer for matrix theory. Amer. Math.
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[21] O. Taussky and J. Todd. A characterisation of algebraic numbers. Proc. Roy.
Irish Acad., Sect. A., 46:1–8, 1940.
[22] O. Taussky and J. Todd. Matrices with finite period. Proc. Edinb. Math.
Soc., II. Ser., 6:128–134, 1940.
[23] O. Taussky and J. Todd. Cholesky, Toeplitz and the triangular factorization
of symmetric matrices. Numer. Algorithms, 41(2):197–202, 2006.
[24] O. Taussky-Todd. Autobiographical Essay. In D. Albers and G. Alexanderson, editors, Mathematical People, Profiles and Interviews. Birkhäuser
Boston Inc., Boston, 1985.
[25] J. Todd. Superpositions of functions. I: Transfinite superpositions of absolutely continuous functions. J. Lond. Math. Soc., 10:166–171, 1935.
[26] J. Todd. Superpositions of functions. II. Transfinite superpositions of absolutely continuous functions. Proc. Lond. Math. Soc., II. Ser., 41:433–439,
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[27] J. Todd. The condition of certain matrices. I. Quart. J. Mech. Appl. Math.,
2:469–472, 1949.
[28] J. Todd. Notes on modern numerical analysis. I. Solution of differential equations by recurrence relations. Math. Tables and Other Aids to Computation,
4:39–44, 1950.
[29] J. Todd. The condition of a certain matrix. Proc. Camb. Philos. Soc., 46:116–
118, 1950.
[30] J. Todd. Experiments on the inversion of a 16 × 16 matrix. Nat. Bur. Standards, Appl. Math. Ser., 29:113–115, 1953.
[31] J. Todd. Experiments in the solution of differential equations by Monte Carlo
methods. J. Washington Acad. Sci., 44:377–381, 1954.
[32] J. Todd. The condition of certain matrices. II. Arch. Math., 5:249–257, 1954.
[33] J. Todd. Motivation for working in numerical analysis. Commun. Pure Appl.
Math., 8:97–116, 1955.
[34] J. Todd. A direct approach to the problem of stability in the numerical
solution of partial differential equations. Commun. Pure Appl. Math., 9:597–
612, 1956.
[35] J. Todd. The condition of certain integral equations. In Symposium on the
numerical treatment of ordinary differential equations, integral and integrodifferential equations (Rome, 1960), pages 306–311. Birkhäuser, Basel, 1960.
[36] J. Todd, editor. Survey of Numerical Analysis. New York-San FranciscoToronto-London: McGraw-Hill Book Company, Inc. XVI, 589 p. , 1962.
[37] J. Todd. Introduction to the Constructive Theory of Functions. Academic
Press Inc., New York, 1963.
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N. MADDEN
[38] J. Todd. Numerical analysis at the National Bureau of Standards. SIAM
Rev., 17:361–370, 1975.
[39] J. Todd. Basic Numerical Mathematics. Vol. 2: Numerical Algebra. International Series of Numerical Mathematics. Vol. 22. Basel - Stuttgart:
Birkhäuser Verlag. 216 p., 1977.
[40] J. Todd. Basic Numerical Mathematics. Vol. 1: Numerical Analysis. International Series of Numerical Mathematics (ISNM), Vol. 14. Basel, Stuttgart:
Birkhäuser Verlag. 253 p., 1979.
[41] J. Todd. Oberwolfach - 1945. In E. F. Beckenback and W. Walter, editors,
General Inequalities 3, 3rd int. Conf., Oberwolfach 1981, ISNM 64, pages
19–22. Birkhäuser Verlag, 1983.
[42] J. Todd. The Weierstrass mean. I: The periods of ℘(z|e1 , e2 , e3 ). Numer.
Math., 57(8):737–746, 1990.
[43] J. Todd and R. S. Varga. Hardy’s inequality and ultrametric matrices. Linear
Algebra Appl., 302/303:33–43, 1999. Special issue dedicated to Hans Schneider (Madison, WI, 1998).
[44] O. Todd. On some boundary value problems in the theory of the non-uniform
supersonic motion of an aerofoil. Rep. and Memoranda no. 2141(8961). Ministry of Supply [London], Aeronaut. Res. Council, 1945.
[45] A. Turing. On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc., II. Ser., 42:230–265, 1936.
[46] A. M. Turing. Rounding-off errors in matrix processes. Quart. J. Mech. Appl.
Math., 1:287–308, 1948.
[47] F. Uhlig. Olga Taussky, a torchbearer for mathematics and teacher of mathematicians. PAMM, 7(1):1170501–1170503, 2007.
[48] R. S. Varga. Geršgorin and his circles, volume 36 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2004.
[49] J. von Neumann and H. H. Goldstine. Numerical inverting of matrices of
high order. Bull. Amer. Math. Soc., 53:1021–1099, 1947.
Niall Madden. Lecturer in Mathematics at NUI Galway. His research interests are in the area of numerical analysis, particularly in developing finite
difference and finite element schemes for solving boundary value problems.
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway
E-mail address: [email protected]